70: Discrete Math and Probability. Programming Computers Superpower! - PowerPoint PPT Presentation

70: Discrete Math and Probability. Programming Computers ≡ Superpower! What are your super powerful programs doing? Logic and Proofs! Induction ≡ Recursion. What can computers do? Work with discrete objects. Discrete Math = ⇒ immense application. Computers learn and interact with the world? E.g. machine learning, data analysis. Probability! See note 1, for more discussion.

Admin. Course Webpage: inst.cs.berkeley.edu/~cs70/sp16 Explains policies, has homework, midterm dates, etc. Two midterms, final. midterm 1 before drop date. (2/16) midterm 2 before grade option change. (3/29) Questions = ⇒ piazza: piazza.com/berkeley/spring2016/cs70 Also: Available after class. Assessment: Two options: Test Only. Midterm 1: 25% Midterm 2: 25% Final: 49% Sundry: 1% Test plus Homework. Test Only Score: 85% Homework Score: 15%

Instructor/Admin Instructors: Satish Rao and Jean Walrand. Both are available throughout the course. Office hours or by email, technical and administrative. Satish Rao: mostly discrete math. Jean Walrand: mostly probability.

Jean ¡Walrand ¡– ¡Prof. ¡of ¡EECS ¡– ¡UCB ¡ 257 ¡Cory ¡Hall ¡– ¡walrand@berkeley.edu ¡ I ¡was ¡born ¡in ¡Belgium (1) ¡and ¡came ¡to ¡Berkeley ¡for ¡my ¡ PhD. ¡ ¡I ¡have ¡been ¡teaching ¡at ¡UCB ¡since ¡1982. ¡ ¡ ¡ My ¡wife ¡and ¡I ¡live ¡in ¡Berkeley. ¡ ¡We ¡have ¡two ¡ daughters ¡(UC ¡alumni ¡– ¡Go ¡Bears!). ¡We ¡like ¡to ¡ski ¡and ¡ play ¡tennis ¡(both ¡poorly). ¡ ¡We ¡enjoy ¡classical ¡music ¡ and ¡jazz. ¡ ¡ ¡ ¡ My ¡research ¡interests ¡include ¡stochasLc ¡systems, ¡ networks ¡and ¡game ¡theory. ¡ (1) ¡

Satish Rao 17th year at Berkeley. PhD: Long time ago, far far away. Research: Theory (Algorithms) Taught: 170, 174, 70, 270, 273, 294, 375, ... Recovering Helicopter(ish) parent of 3 College(ish) kids.

Wason’s experiment:1 Suppose we have four cards on a table: ◮ 1st about Alice, 2nd about Bob, 3rd Charlie, 4th Donna. ◮ Card contains person’s destination on one side, and mode of travel. ◮ Consider the theory: “If a person travels to Chicago, he/she flies.” ◮ Suppose you see that Alice went to Baltimore, Bob drove, Charlie went to Chicago, and Donna flew. Alice Bob Charlie Donna Chicago Baltimore drove flew ◮ Which cards do you need to flip to test the theory? Answer: Later.

CS70: Lecture 1. Outline. Today: Note 1. Note 0 is background. Do read/skim it. The language of proofs! 1. Propositions. 2. Propositional Forms. 3. Implication. 4. Truth Tables 5. Quantifiers 6. More De Morgan’s Laws

Propositions: Statements that are true or false. √ 2 is irrational Proposition True 2+2 = 4 Proposition True 2+2 = 3 Proposition False 826th digit of pi is 4 Proposition False Johny Depp is a good actor Not a Proposition All evens > 2 are sums of 2 primes Proposition False 4 + 5 Not a Proposition. x + x Not a Proposition. Alice travelled to Chicago Proposition. False Again: “value” of a proposition is ... True or False

Propositional Forms. Put propositions together to make another... Conjunction (“and”): P ∧ Q “ P ∧ Q ” is True when both P and Q are True . Else False . Disjunction (“or”): P ∨ Q “ P ∨ Q ” is True when at least one P or Q is True . Else False . Negation (“not”): ¬ P “ ¬ P ” is True when P is False . Else False . Examples: ¬ “ ( 2 + 2 = 4 ) ” – a proposition that is ... False “2 + 2 = 3” ∧ “2 + 2 = 4” – a proposition that is ... False “2 + 2 = 3” ∨ “2 + 2 = 4” – a proposition that is ... True

Propositional Forms: quick check! √ P = “ 2 is rational” Q = “826th digit of pi is 2” P is ...False . Q is ...True . P ∧ Q ... False P ∨ Q ... True ¬ P ... True

Put them together.. Propositions: P 1 - Person 1 rides the bus. P 2 - Person 2 rides the bus. .... But we can’t have either of the following happen; That either person 1 or person 2 ride the bus and person 3 or 4 ride the bus. Or that person 2 or person 3 ride the bus and that either person 4 ride the bus or person 5 doesn’t. Propositional Form: ¬ ((( P 1 ∨ P 2 ) ∧ ( P 3 ∨ P 4 )) ∨ (( P 2 ∨ P 3 ) ∧ ( P 4 ∨¬ P 5 ))) Can person 3 ride the bus? Can person 3 and person 4 ride the bus together? This seems ...complicated. We can program!!!! We need a way to keep track!

Truth Tables for Propositional Forms. P Q P ∧ Q P Q P ∨ Q T T T T T T T F F T F T F T F F T T F F F F F F Notice: ∧ and ∨ are commutative. One use for truth tables: Logical Equivalence of propositional forms! Example: ¬ ( P ∧ Q ) logically equivalent to ¬ P ∨¬ Q ...because the two propositional forms have the same... ....Truth Table! P Q ¬ ( P ∨ Q ) ¬ P ∧¬ Q T T F F T F F F F T F F F F T T DeMorgan’s Law’s for Negation: distribute and flip! ¬ ( P ∧ Q ) ≡ ¬ P ∨¬ Q ¬ ( P ∨ Q ) ≡ ¬ P ∧¬ Q

Distributive? P ∧ ( Q ∨ R ) ≡ ( P ∧ Q ) ∨ ( P ∧ R ) ? Simplify: ( T ∧ Q ) ≡ Q , ( F ∧ Q ) ≡ F . Cases: P is True . LHS: T ∧ ( Q ∨ R ) ≡ ( Q ∨ R ) . RHS: ( T ∧ Q ) ∨ ( T ∧ R ) ≡ ( Q ∨ R ) . P is False . LHS: F ∧ ( Q ∨ R ) ≡ F . RHS: ( F ∧ Q ) ∨ ( F ∧ R ) ≡ ( F ∨ F ) ≡ F . P ∨ ( Q ∧ R ) ≡ ( P ∨ Q ) ∧ ( P ∨ R ) ? Simplify: T ∨ Q ≡ T , F ∨ Q ≡ Q . Foil 1: ( A ∨ B ) ∧ ( C ∨ D ) ≡ ( A ∧ C ) ∨ ( A ∧ D ) ∨ ( B ∧ C ) ∨ ( B ∧ D ) ? Foil 2: ( A ∧ B ) ∨ ( C ∧ D ) ≡ ( A ∨ C ) ∧ ( A ∨ D ) ∧ ( B ∨ C ) ∧ ( B ∨ D ) ?

Implication. P = ⇒ Q interpreted as If P , then Q . True Statements: P , P = ⇒ Q . Conclude: Q is true. Examples: Statement: If you stand in the rain, then you’ll get wet. P = “you stand in the rain” Q = “you will get wet” Statement: “Stand in the rain” Can conclude: “you’ll get wet.” Statement: If a right triangle has sidelengths a ≤ b ≤ c , then a 2 + b 2 = c 2 . P = “a right triangle has sidelengths a ≤ b ≤ c ”, Q = “ a 2 + b 2 = c 2 ”.

Non-Consequences/consequences of Implication The statement “ P = ⇒ Q ” only is False if P is True and Q is False . False implies nothing P False means Q can be True or False Anything implies true. P can be True or False when Q is True If chemical plant pollutes river, fish die. If fish die, did chemical plant pollute river? Not necessarily. P = ⇒ Q and Q are True does not mean P is True Be careful! Instead we have: P = ⇒ Q and P are True does mean Q is True . The chemical plant pollutes river. Can we conclude fish die? Some Fun: use propositional formulas to describe implication? (( P = ⇒ Q ) ∧ P ) = ⇒ Q .

Implication and English. P = ⇒ Q ◮ If P , then Q . ◮ Q if P . Just reversing the order. ◮ P only if Q . Remember if P is true then Q must be true. this suggests that P can only be true if Q is true. since if Q is false P must have been false. ◮ P is sufficient for Q . This means that proving P allows you to conclude that Q is true. ◮ Q is necessary for P . For P to be true it is necessary that Q is true. Or if Q is false then we know that P is false.

Truth Table: implication. P Q P = ⇒ Q P Q ¬ P ∨ Q T T T T T T T F F T F F F T T F T T F F T F F T ¬ P ∨ Q ≡ P = ⇒ Q . These two propositional forms are logically equivalent!

Contrapositive, Converse ◮ Contrapositive of P = ⇒ Q is ¬ Q = ⇒ ¬ P . ◮ If the plant pollutes, fish die. ◮ If the fish don’t die, the plant does not pollute. (contrapositive) ◮ If you stand in the rain, you get wet. ◮ If you did not stand in the rain, you did not get wet. (not contrapositive!) converse! ◮ If you did not get wet, you did not stand in the rain. (contrapositive.) Logically equivalent! Notation: ≡ . P = ⇒ Q ≡ ¬ P ∨ Q ≡ ¬ ( ¬ Q ) ∨¬ P ≡ ¬ Q = ⇒ ¬ P . ◮ Converse of P = ⇒ Q is Q = ⇒ P . If fish die the plant pollutes. Not logically equivalent! ◮ Definition: If P = ⇒ Q and Q = ⇒ P is P if and only if Q or P ⇐ ⇒ Q . (Logically Equivalent: ⇐ ⇒ . )

Variables. Propositions? ◮ ∑ n i = 1 i = n ( n + 1 ) . 2 ◮ x > 2 ◮ n is even and the sum of two primes No. They have a free variable. We call them predicates, e.g., Q ( x ) = “ x is even” Same as boolean valued functions from 61A or 61AS! i = 1 i = n ( n + 1 ) ◮ P ( n ) = “ ∑ n . ” 2 ◮ R ( x ) = “ x > 2” ◮ G ( n ) = “ n is even and the sum of two primes” ◮ Remember Wason’s experiment! F ( x ) = “Person x flew.” C ( x ) = “Person x went to Chicago ◮ C ( x ) = ⇒ F ( x ) . Theory from Wason’s. If person x goes to Chicago then person x flew. Next: Statements about boolean valued functions!!

Quantifiers.. There exists quantifier: ( ∃ x ∈ S )( P ( x )) means ”There exists an x in S where P ( x ) is true.” For example: ( ∃ x ∈ N )( x = x 2 ) Equivalent to “ ( 0 = 0 ) ∨ ( 1 = 1 ) ∨ ( 2 = 4 ) ∨ ... ” Much shorter to use a quantifier! For all quantifier; ( ∀ x ∈ S ) ( P ( x )) . means “For all x in S , we have P ( x ) is True .” Examples: “Adding 1 makes a bigger number.” ( ∀ x ∈ N ) ( x + 1 > x ) ”the square of a number is always non-negative” ( ∀ x ∈ N )( x 2 > = 0 ) Wait! What is N ?